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Full text of "Mathematical And Physical Papers - Iii"

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The first of the equations of condition (117) requires that
D=0,    C = -iF .....................(120).
It is particularly to be remarked that inasmuch as the two arbitrary constants C, D are determined by the first of the conditions (117), none remain whereby to satisfy the second. Nevertheless it happens that the second of these conditions leads to precisely the same equations (120) as the first. The equations of condition
(116) give
A = -\Va\    B^lVa;
2r* . 3r
If now we wish to obtain the solution of the problem in its original shape, in which the sphere is in motion and the fluid at rest, except so far as it is disturbed by the sphere, we have merely to add Fcos#,  Fsin#, J Fr2sin2# to the expressions for R, , ^. We get from (121)
/Q        \
40. Let us now return to the problem of Section II. ; let us suppose the time of oscillation to increase indefinitely, and examine what equation (40) becomes in the limit.
When r becomes infinite, n, and therefore m, vanishes; the expression within brackets in (40) takes the form oo  oo , and its limiting value is easily found by the ordinary methods. We must retain the m2 in the coefficient of t, because t is susceptible of unlimited increase. We get in the limit
(125). ^      '
* I have already had occasion, in treating of another subject, to publish the solution expressed by this equation, which I had obtained as a limiting case of the problem of a ball pendulum. See Philosophical Magazine for May, 1848, p. 343. [Ante, Vol. n. p. 10.]